A Generative Theory of Shape

(Springer-Verlag, 550pages). The purpose of the book is to develop a
generative theory of shape that has two properties regarded as fundamental
to intelligence - maximizing transfer of structure and maximizing
recoverability of the generative operations. These two properties are
particularly important in the representation of complex shape - which is the
main concern of the book. The primary goal of the theory is the conversion
of complexity into understandability. For this purpose, a mathematical
theory is presented of how understandability is created in a structure. This
is achieved by developing a group-theoretic approach to formalizing transfer
and recoverability. To handle complex shape, a new class of groups is
developed, called unfolding groups. These unfold structure from a maximally
collapsed version of that structure. A principal aspect of the theory is
that it develops a group-theoretic formalization of major object-oriented
concepts such as inheritance. The result is an object-oriented theory of
geometry.

The algebraic theory is applied in detail to CAD, perception, and robotics.
In CAD, lengthy chapters are presented on mechanical and architectural
design. For example, using the theory of unfolding groups, the book works in
detail through the main stages of mechanical CAD/CAM: part-design, assembly
and machining. And within part-design, an extensive algebraic analysis is
given of sketching, alignment, dimensioning, resolution, editing, sweeping,
feature-addition, and intent-management. The equivalent analysis is also
done for architectural design. In perception, extensive theories are given
for grouping and the main Gestalt motion phenomena (induced motion,
separation of systems, the Johannson relative/absolute motion effects); as
well as orientation and form. In robotics, several levels of analysis are
developed for manipulator structure, using the author's algebraic theory of
object-oriented structure.